N invitations in Rashida's Birthdays

Using the pigeonhole principle, you can prove that there must be a string of consecutive days on which exactly $7$ invitations were issued.

Let $a_i$ be the cumulative number of invitations sent up to and including day $i$. Then $\{a_1, a_2, \ldots, a_{29}, a_1+7, a_2+7, \ldots, a_{29}+7 \}$ is a set of $58$ integers between $1$ and $57$, inclusive. Therefore, two of the integers, $a_i$ and $a_j+7$, must be the same. (Since at least one invitation was issued on each day, $i \neq j \Rightarrow a_i \neq a_j \Rightarrow a_i+7 \neq a_j+7$.) That means that exactly $7$ invitations must have been issued between days $j+1$ and $i$, inclusive.